Abstract

The formation of localized states or modes at defects in waveguide arrays is investigated both, theoretically and experimentally. If the effective index or the coupling of the defect guide to its neighbors is varied the number and character of respective modes bound to the defect can be altered. Waveguide arrays may be considered as tailor-made or metamaterials with new and unexpected properties as e.g. guiding staggered modes bound to defects with reduced index. Although the symmetric defect waveguide becomes multimode for increased coupling it does not support antisymmetric modes. All theoretical predictions are confirmed in excellent agreement with experimental observations in polymer waveguide arrays.

©2003 Optical Society of America

With the advance of optical data transfer the requirements on quality, reliability and functionality of integrated optics devices have increased. However, the sole use of conventional materials implies severe constraints on the performance of optical elements. Mature fabrication technologies have now allowed manufacturing tailor-made nanostructured materials with optimized properties (metamaterials). Photonic crystals as periodic sub-wavelength structures are probably the most prominent example for a metamaterial where e.g. refraction and diffraction can be varied to a large extent [1]. Thus light propagation may be strongly effected and even controlled [2,3]. Waveguide arrays are simpler but also promising candidates, where light propagation can be considerably modified compared with that in bulk materials. In the array waves are confined to individual guides. The evolution of this virtually “discretized” light is restricted to energy transfer via the evanescent tails of respective guided modes. The gap between photonic crystals and waveguide arrays is bridged by photonic crystal fibers, which are periodic in transverse direction, but translationally invariant with respect to the direction of propagation. Hence, some of the effects, which we investigate in waveguide arrays can likewise be observed in photonic crystal fibers.

Currently linear and nonlinear dynamics in discrete optical systems as waveguide arrays are subject of active research. Discrete diffraction and refraction in homogeneous arrays was demonstrated to deviate considerably from that in bulk materials [4,5]. First investigations of inhomogeneous waveguide arrays (index gradient) disclosed that the optical field performs photonic Bloch-oscillations across the array [6,7]. Hence the waveguide array itself can be regarded as an artificial, tailor-made material with new, peculiar properties. In particular it is worthwhile to study how arrays perform as basic materials of waveguide optics and how modes can be guided in these periodic structures. Where light spreads in homogeneous arrays, it is reflected [8] or trapped [9] by inhomogeneities. Those defect modes can have new and exciting properties as it was demonstrated for photonic crystal fibers. Single mode operation was obtained in a huge wavelength domain [10], extremely small [11] or large [12] effective mode areas were achieved and almost arbitrary values of the group velocity dispersion [13] can be reached. The aim of this paper is to investigate theoretically as well as experimentally basic features of defect modes in waveguide arrays where we will predict and confirm areas of existence of various types of guided modes.

The investigated arrays consist of 101 waveguides of an inorganic-organic hybrid polymer [14] (nco=1.547 @ 633nm) on thermally oxidized silicon wafers (nsub=1.457 @ 633nm) with a polymer cladding (ncl=1.544 @ 633nm) (see Fig. 1). The samples are fabricated by UV lithography [15] on 4 inch wafers leading to propagation lengths up to 7cm.

 figure: Fig. 1.

Fig. 1. Cross section of a polymer waveguide array with a defect. The defect is introduced by reducing the waveguide spacing compared with the homogenous array.

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All waveguides have the same height of 3.5µm. Waveguide widths between 2.5 and 4.5µm provide low loss single mode wave-guiding (<0.04dB/cm @ 633 nm) and a waveguide spacing between 4 and 5µm ensures an efficient evanescent coupling of the nearest neighboring guides. Defects and inhomogeneities are easily incorporated by locally varying either the width of an individual guide or the spacing around it.

Before presenting the experimental results let us give some theoretical estimates of the effects, which we expect to observe. Because coupling is weak we can apply a coupled-mode theory, where the evolution of the modal amplitude An in the nth single mode waveguide is described by:

(iZ+β0)An+C(An+1+An1)=0,

where β0 is the propagation constant of the individual guide and C denotes the coupling constant between neighboring waveguides. Eigensolutions of Eq. (1) are plane waves or Bloch-modes of the form An=A·exp(iβZ-iκn), where κ is the normalized Bloch vector or the phase difference between adjacent guides corresponding to a tilt. Inserting this ansatz into Eq. (1) we find β to be entirely defined by κ giving the so-called diffraction relation (Fig. 2(a))

β=β0+2Ccos(κ).

In contrast to bulk media the range of propagation constants β of freely propagating waves is limited. Its width depends on the coupling constant C and its position is defined by the wavenumber of the individual guides β0. Outside this band only waves exist, which decay exponentially in transverse direction.

 figure: Fig. 2.

Fig. 2. Dispersion relation of Bloch modes and the formation of defect modes (a) diffraction relation of Bloch-waves (longitudinal vs. transverse wavenumber) in a homogeneous waveguide array (propagation constant of isolated waveguides: β0, coupling constant C). In the shaded regions only evanescent waves exist. (b) Shift of the band structure and formation of a staggered mode (wavenumber βD) around a defect with a wavenumber reduced by δβ. (c) Expansion of the band structure and formation of staggered and unstaggered modes around a defect with increased coupling (C 1>C) (d) Compression of the band structure around a defect with reduced coupling (C 1<C).

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Hence, any guided mode must have a propagation constant, which is situated outside the band defined by Eq. (2). But, a localized mode also requires an inhomogeneity to exist. Because we restrict ourselves to a symmetric defect at n=0 respective coupled mode equations of the fields at n=0,±1 are modified accordingly as (iZ+β0+δβ)A0+C1(A+1+A1)=0 and

(iZ+β0)A±1+C1A0+CA±2=0,

where C 1 is the modified coupling constant and δβ the change in the propagation constant of the defect guide. Any mode bound to the defect has the form

An(Z)=anexp(iβDZ)

with a fixed field shape an. To determine whether the defect can indeed carry a guided mode we have to perform some mathematics. Because exponentially decaying tails are required, we assume the field shapes to obey

a±n=a±1γn1forn2

where

γ<1

holds. Inserting the ansatz (5) into Eq. (1) we immediately obtain the propagation constant of guided modes as a continuation of the diffraction relation (2) as

βD=β0+C(γ+1γ)

Because we restrict ourselves to symmetric defects respective guided modes must be either symmetric or antisymmetric. For an antisymmetric mode (a+1=-a-1) the field amplitude at guide n=0 must vanish (a 0=0) for symmetry reasons. Hence all changes induced by the defect in Eq. (3) have no effect and the defect itself becomes invisible for antisymmetric modes. Consequently no field is bound and no guided mode with odd symmetry exists. Therefore only symmetric modes have to be considered. Assuming a+1=a-1 and inserting Eqs. (4), (5) and (7) into Eq. (3) we end up with an eigenvalue problem for the transverse decay rate of the field structure γ as

1γ=δβ2C±(δβ2C)2+2(C1C)21.

If (C1/C)2>1-δβ/2C holds γ is positive and fulfills inequality (6), i.e., 0<γ<1. The respective guided mode is called “unstaggered” because it possesses a flat phase (see Fig. 3(a)). Following Eq. (7) its propagation constant lies above the band of Bloch states of the homogenous array. However γ can also be negative if the condition (C1/C)2>1+δβ/2C is fulfilled giving rise to the formation of a staggered mode with a phase difference π between adjacent waveguides (see Fig. 3(c)). In contrast to guided modes in conventional materials the wavenumber of the respective guided mode is below those of the continuous states in the homogenous array.

 figure: Fig. 3.

Fig. 3. Field and intensity of a staggered and an unstaggered mode for a dominant change of the propagation constant of the defect. Field (a) and intensity (b) distribution of an unstaggered defect mode for δβ/C=2.0 and C 1/C=0.8 (cross 1 in Fig. 4), solid line: theory, dots: experiment. Field (c) and intensity (d) distribution of a staggered defect mode for δβ/C=-1.4 and C 1/C=1.1 (cross 2 in fig. 4), solid line: theory, dots: experiment.

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Hence in the parameter space defined by C 1 and δβ we can find areas without guided modes, with single mode waveguiding, either a staggered or an unstaggered mode, or domains with both types of modes coexisting (see Fig. 4).

 figure: Fig. 4.

Fig. 4. Regions of existence for symmetric staggered and unstaggered modes in the (C1/C)2-δβ-plane. The crosses mark the parameters for the experiments (Figs. 3, 5 and 6).

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To a certain extent waveguiding in conventional materials is reproduced. For instance we find an unstaggered mode, if only the refractive index of the defect guide is increased compared to the homogeneous array: δβ>0. However, contrary to waveguiding in homogeneous media a localized state also appears in form of a staggered mode, if the wavenumber of the central guide is decreased: δβ<0.

To understand the peculiarities of waveguide arrays it is useful to assume that the defect itself forms its own tiny array with an individual band structure extending between the boundaries β0+δβ±2C1. As soon as this “defect band” extends further than that of the homogeneous array, a localized state can be formed. The respective wavenumber βD must be contained in the “defect band” but missed in the band of the homogeneous array. This simple picture also explains that if the coupling around the defect is increased C1>C we find both staggered and unstaggered modes to appear (see Fig. 2(c)) on both sides of the band of the homogeneous array. In contrast to that, no guidance is observed for a decreased coupling C1<C. In this case all wavenumbers of the defect band are phase matched to waves of the homogenous array (see Fig. 2(d)).

Obviously this simple picture fails to explain the guiding properties accurately in particular, for simultaneous variations of wavenumber δβ and coupling C 1 of the defect. The main reason is the lack of a continuous density of states in the “defect array”. If the number of waveguides in the “defect array” would be infinitely large the above method to determine guided defect modes would become exact.

We are now going to compare the theoretical predictions with experimental results. Keeping in mind that the properties of a mode guided by the defect depend solely on the two parameters δβ/C and C 1/C we fabricated samples to test defects belonging to the four distinct areas in the parameter plane (see Fig. 4). To this end we varied width and spacing of the investigated defects. Where a modification of the waveguide spacing solely influences the coupling between the neighboring waveguides, a variation of the width of a guide does not only change the wavenumber of the defect but alters the coupling as well.

Compared to these deliberately induced perturbations conventional fabrication tolerances were much smaller and originate almost exclusively from a variation of the waveguide layer thickness due to the spin coating process. The spatial scale of resulting inhomogenities is comparable with the width of the wafer. Hence, our structures are mainly subject to a constant drift of parameters. Comparing samples originating from different parts of the wafer we found the strength of the coupling to change by approximately 5%. Within one sample no transverse variations of the array parameters could be observed.

A single waveguide excitation was implemented by focusing a HeNe laser beam on the entrance facet with a microscope objective. The light emitted from the end faced was detected by a CCD-camera.

First we checked the existence of an unstaggered mode by creating a single guide with increased width of 3.5µm compared with 3µm in the homogeneous part of the array. As a consequence the propagation constant of the defect guide is increased (δβ/C=2). But additionally the coupling is slightly decreased C 1/C=0.8 (see cross 1 in Fig. 4). Similar to conventional waveguiding light concentrates around the region of higher effective index and the modal fields have a flat phase (see Fig. 3(b)).

Next we investigated deviations from classical waveguiding mechanisms. Hence we looked for a staggered mode by decreasing the width of the defect waveguide (3µm compared with 3.5µm in the remaining array). Again the resulting decrease of the propagation constant of the defect (δβ/C=-1.4) is accompanied by a small increase of the coupling constant (C 1/C=1.1, see cross 2 in Fig. 4). In fact we also observed a guided mode (see Fig. 3(d)), whose shape differs considerably from that of an unstaggered one. Because fields in adjacent guides are π out of phase, the intensity of a staggered mode becomes zero between the waveguides due to destructive interference of respective modal fields. Hence, in contrast to the unstaggered mode, which is bound by internal reflection, the guiding mechanism of the staggered state relies on Bragg reflection on the periodic structure of the array.

In case of a dominant change of the coupling constant C 1, two different regions occur in the δβ/C-(C 1/C)2 -plane. For an increase of the defect coupling C 1>C both, unstaggered and staggered modes exist. A nearly exclusive increase of the coupling constant C 1 can be experimentally achieved by decreasing the spacing between the center waveguide and its neighbors (spacing: 4µm compared with 5µm in the rest of the array). The corresponding parameters are C 1/C=1.4 and δβ/C=-0.3 (see cross 3 in Fig. 4). An input beam centered on a single waveguide always excites both modes. At the end facet of the array an interference pattern is observed depending on the actual phase difference between the two bound states. Since both modes have different propagation constants their phase relation changes on propagation. More importantly, already the initial phase difference depends on the point of excitation. If the exciting beam is shifted from the defect guide (n=0) towards its neighbor (n=±1) the phase difference between the staggered and unstaggered modes changes by π. Hence, by varying the waveguide of excitation we can switch between destructive and constructive interference in e.g. the defect guide at the output facet (compare Figs. 5(a) and (b)). Because the phase of the staggered mode alternates whereas that of the unstaggered one remains flat a constructive interference of both modes on the defect site is accompanied by destructive interference in the neighboring site and vice versa. Hence we either observe a maximum in guide n=0 or n=±1. Even if the initial excitation is asymmetric with respect to the defect guide we never observe an asymmetric guided field at the output. Hence, as predicted no asymmetric mode exists, although the defect is multimode.

 figure: Fig. 5.

Fig. 5. Interference pattern of a staggered and an unstaggered defect mode for dominant change of the coupling constant (δβ/C=-0.3 and C 1/C=1.4, cross 3 in Fig. 4) of the defect at a propagation distance of 59,95mm. Dots: experiment, lines: theory, dashed line: position of the excitation. (b) Intensity distribution for an excitation of the defect waveguide. (a) and (c) Intensity distribution for an excitation of the left and right nearest neighbor waveguide of the defect. Insets: schematic diagrams of the modal amplitude of the unstaggered and staggered mode, the superposition of both modal fields produces the actual interference pattern.

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The analytical theory predicts that there are no bound states if the coupling constant of the defect waveguide is decreased (C 1<C - cross 4 in Fig. 4). This is somehow contra intuitive. Because the defect tends to be isolated due to the reduced coupling one would even expect improved guiding properties. Again the simplified model of the local band around the defect helps to explain the effect (see Fig. 2(d)). A decrease of the coupling constant results in a band shrinkage. Hence all states of the defect band are phase matched to those of the homogenous array. Light from the defect predominantly couples into Bloch modes of the middle of the band, which have a high transverse velocity. Hence, the excitation will leave a defect with reduced coupling very quickly as demonstrated in the experiment (see Fig. 6(a) and (b)). In contrast to an excitation in the homogenous array, where parts of the field also propagate straight (see Fig. 6(c) and (d)), the defect repels the light causing a dark region around it.

 figure: Fig. 6.

Fig. 6. Diffraction pattern for an excitation of a repulsive defect ((a) theory, (b) experiment) with reduced coupling (C 1/C=0.5, δβ/C=0, cross 4 in Fig.4) and in a homogeneous array ((c) theory, (d) experiment).

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To conclude, we have analytically determined and experimentally verified the range of existence of localized defect states or guided modes in a waveguide array. Both, an increase of the coupling constant as well as the variation of the effective index of the defect guide give rise to the formation of localized states. Staggered modes, which are not known from conventional materials are found. Furthermore it turns out that symmetric defects in waveguide arrays cannot support antisymmetric modes. This is to our knowledge the only optical system, which although multimode, supports symmetric modes only.

References and links

1. Focus Issue: “Negative Refraction and Metamaterials,” ed. by J. B. Pendry, Opt. Express 11, 639 (2003), http://www.opticsexpress.org/issue.cfm?issue_id=186 [CrossRef]   [PubMed]  

2. M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000). [CrossRef]  

3. G. von Freymanna, W. Koch, D. C. Meisel, and M. Wegener, “Diffraction properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 83, 614–616 (2003). [CrossRef]  

4. S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–48 (1973). [CrossRef]  

5. T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous Refraction and Diffraction in Discrete Optical Systems,” Phys. Rev. Lett. 88, 093901–093904 (2002). [CrossRef]   [PubMed]  

6. T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999). [CrossRef]  

7. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999). [CrossRef]  

8. R. Morandotti, H. S. Eisenberg, D. Mandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, and J. S. Aitchison, “Interactions of discrete solitons with structural defects,” Opt. Lett. 28, 834–836 (2003). [CrossRef]   [PubMed]  

9. U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Nonlinearly Induced Escape from a Defect State,” Appl. Phys. Lett. 75, 1384–1386 (1999). [CrossRef]  

10. T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997). [CrossRef]   [PubMed]  

11. P Russell, “Photonic Crystal Fibers,” Science 299, 358–363 (2003). [CrossRef]   [PubMed]  

12. J. C. Knight, T. A. Birks, R. F. Cregan, P. ST. J. Russell, and J.-P. de Sandro, “Large Mode Area Photonic Crystal Fiber,” Electron. Lett. 34, 1347–1348 (1998). [CrossRef]  

13. D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Group-velocity dispersionin photonic crystal fibers,” Opt. Lett. 23, 1662–1664 (1998). [CrossRef]  

14. R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J.-U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic-organic hybrid materials for application in optical devices,” Thin Solid Films 422, 194–200 (2003) [CrossRef]  

15. U. Streppel, P. Dannberg, C. Wächter, A. Bräuer, L. Fröhlich, R. Houbertz, and M. Popall, “New wafer-scale fabrication method for stacked optical waveguide interconnects and 3D micro-optic structures using photo-responsive (inorganic-organic hybrid) polymers,” Opt. Mat. 21, 475–483 (2002). [CrossRef]  

References

  • View by:

  1. Focus Issue: “Negative Refraction and Metamaterials,” ed. by J. B. Pendry, Opt. Express 11, 639 (2003), http://www.opticsexpress.org/issue.cfm?issue_id=186
    [Crossref] [PubMed]
  2. M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
    [Crossref]
  3. G. von Freymanna, W. Koch, D. C. Meisel, and M. Wegener, “Diffraction properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 83, 614–616 (2003).
    [Crossref]
  4. S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–48 (1973).
    [Crossref]
  5. T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous Refraction and Diffraction in Discrete Optical Systems,” Phys. Rev. Lett. 88, 093901–093904 (2002).
    [Crossref] [PubMed]
  6. T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
    [Crossref]
  7. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
    [Crossref]
  8. R. Morandotti, H. S. Eisenberg, D. Mandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, and J. S. Aitchison, “Interactions of discrete solitons with structural defects,” Opt. Lett. 28, 834–836 (2003).
    [Crossref] [PubMed]
  9. U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Nonlinearly Induced Escape from a Defect State,” Appl. Phys. Lett. 75, 1384–1386 (1999).
    [Crossref]
  10. T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997).
    [Crossref] [PubMed]
  11. P Russell, “Photonic Crystal Fibers,” Science 299, 358–363 (2003).
    [Crossref] [PubMed]
  12. J. C. Knight, T. A. Birks, R. F. Cregan, P. ST. J. Russell, and J.-P. de Sandro, “Large Mode Area Photonic Crystal Fiber,” Electron. Lett. 34, 1347–1348 (1998).
    [Crossref]
  13. D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Group-velocity dispersionin photonic crystal fibers,” Opt. Lett. 23, 1662–1664 (1998).
    [Crossref]
  14. R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J.-U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic-organic hybrid materials for application in optical devices,” Thin Solid Films 422, 194–200 (2003)
    [Crossref]
  15. U. Streppel, P. Dannberg, C. Wächter, A. Bräuer, L. Fröhlich, R. Houbertz, and M. Popall, “New wafer-scale fabrication method for stacked optical waveguide interconnects and 3D micro-optic structures using photo-responsive (inorganic-organic hybrid) polymers,” Opt. Mat. 21, 475–483 (2002).
    [Crossref]

2003 (5)

Focus Issue: “Negative Refraction and Metamaterials,” ed. by J. B. Pendry, Opt. Express 11, 639 (2003), http://www.opticsexpress.org/issue.cfm?issue_id=186
[Crossref] [PubMed]

G. von Freymanna, W. Koch, D. C. Meisel, and M. Wegener, “Diffraction properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 83, 614–616 (2003).
[Crossref]

R. Morandotti, H. S. Eisenberg, D. Mandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, and J. S. Aitchison, “Interactions of discrete solitons with structural defects,” Opt. Lett. 28, 834–836 (2003).
[Crossref] [PubMed]

P Russell, “Photonic Crystal Fibers,” Science 299, 358–363 (2003).
[Crossref] [PubMed]

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J.-U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic-organic hybrid materials for application in optical devices,” Thin Solid Films 422, 194–200 (2003)
[Crossref]

2002 (2)

U. Streppel, P. Dannberg, C. Wächter, A. Bräuer, L. Fröhlich, R. Houbertz, and M. Popall, “New wafer-scale fabrication method for stacked optical waveguide interconnects and 3D micro-optic structures using photo-responsive (inorganic-organic hybrid) polymers,” Opt. Mat. 21, 475–483 (2002).
[Crossref]

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous Refraction and Diffraction in Discrete Optical Systems,” Phys. Rev. Lett. 88, 093901–093904 (2002).
[Crossref] [PubMed]

2000 (1)

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
[Crossref]

1999 (3)

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[Crossref]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[Crossref]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Nonlinearly Induced Escape from a Defect State,” Appl. Phys. Lett. 75, 1384–1386 (1999).
[Crossref]

1998 (2)

J. C. Knight, T. A. Birks, R. F. Cregan, P. ST. J. Russell, and J.-P. de Sandro, “Large Mode Area Photonic Crystal Fiber,” Electron. Lett. 34, 1347–1348 (1998).
[Crossref]

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Group-velocity dispersionin photonic crystal fibers,” Opt. Lett. 23, 1662–1664 (1998).
[Crossref]

1997 (1)

1973 (1)

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–48 (1973).
[Crossref]

Aitchison, J. S.

R. Morandotti, H. S. Eisenberg, D. Mandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, and J. S. Aitchison, “Interactions of discrete solitons with structural defects,” Opt. Lett. 28, 834–836 (2003).
[Crossref] [PubMed]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Nonlinearly Induced Escape from a Defect State,” Appl. Phys. Lett. 75, 1384–1386 (1999).
[Crossref]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[Crossref]

Birks, T. A.

Bräuer, A.

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J.-U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic-organic hybrid materials for application in optical devices,” Thin Solid Films 422, 194–200 (2003)
[Crossref]

U. Streppel, P. Dannberg, C. Wächter, A. Bräuer, L. Fröhlich, R. Houbertz, and M. Popall, “New wafer-scale fabrication method for stacked optical waveguide interconnects and 3D micro-optic structures using photo-responsive (inorganic-organic hybrid) polymers,” Opt. Mat. 21, 475–483 (2002).
[Crossref]

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous Refraction and Diffraction in Discrete Optical Systems,” Phys. Rev. Lett. 88, 093901–093904 (2002).
[Crossref] [PubMed]

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[Crossref]

Buestrich, R.

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J.-U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic-organic hybrid materials for application in optical devices,” Thin Solid Films 422, 194–200 (2003)
[Crossref]

Cregan, R. F.

J. C. Knight, T. A. Birks, R. F. Cregan, P. ST. J. Russell, and J.-P. de Sandro, “Large Mode Area Photonic Crystal Fiber,” Electron. Lett. 34, 1347–1348 (1998).
[Crossref]

Cronauer, C.

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J.-U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic-organic hybrid materials for application in optical devices,” Thin Solid Films 422, 194–200 (2003)
[Crossref]

Dannberg, P.

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J.-U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic-organic hybrid materials for application in optical devices,” Thin Solid Films 422, 194–200 (2003)
[Crossref]

U. Streppel, P. Dannberg, C. Wächter, A. Bräuer, L. Fröhlich, R. Houbertz, and M. Popall, “New wafer-scale fabrication method for stacked optical waveguide interconnects and 3D micro-optic structures using photo-responsive (inorganic-organic hybrid) polymers,” Opt. Mat. 21, 475–483 (2002).
[Crossref]

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[Crossref]

de Sandro, J.-P.

J. C. Knight, T. A. Birks, R. F. Cregan, P. ST. J. Russell, and J.-P. de Sandro, “Large Mode Area Photonic Crystal Fiber,” Electron. Lett. 34, 1347–1348 (1998).
[Crossref]

Domann, G.

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J.-U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic-organic hybrid materials for application in optical devices,” Thin Solid Films 422, 194–200 (2003)
[Crossref]

Eisenberg, H. S.

R. Morandotti, H. S. Eisenberg, D. Mandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, and J. S. Aitchison, “Interactions of discrete solitons with structural defects,” Opt. Lett. 28, 834–836 (2003).
[Crossref] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[Crossref]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Nonlinearly Induced Escape from a Defect State,” Appl. Phys. Lett. 75, 1384–1386 (1999).
[Crossref]

Elflein, W.

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[Crossref]

Fröhlich, L.

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J.-U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic-organic hybrid materials for application in optical devices,” Thin Solid Films 422, 194–200 (2003)
[Crossref]

U. Streppel, P. Dannberg, C. Wächter, A. Bräuer, L. Fröhlich, R. Houbertz, and M. Popall, “New wafer-scale fabrication method for stacked optical waveguide interconnects and 3D micro-optic structures using photo-responsive (inorganic-organic hybrid) polymers,” Opt. Mat. 21, 475–483 (2002).
[Crossref]

Garmire, E.

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–48 (1973).
[Crossref]

Garvin, H. L.

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–48 (1973).
[Crossref]

Houbertz, R.

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J.-U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic-organic hybrid materials for application in optical devices,” Thin Solid Films 422, 194–200 (2003)
[Crossref]

U. Streppel, P. Dannberg, C. Wächter, A. Bräuer, L. Fröhlich, R. Houbertz, and M. Popall, “New wafer-scale fabrication method for stacked optical waveguide interconnects and 3D micro-optic structures using photo-responsive (inorganic-organic hybrid) polymers,” Opt. Mat. 21, 475–483 (2002).
[Crossref]

Hunsperger, R. G.

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–48 (1973).
[Crossref]

Knight, J. C.

J. C. Knight, T. A. Birks, R. F. Cregan, P. ST. J. Russell, and J.-P. de Sandro, “Large Mode Area Photonic Crystal Fiber,” Electron. Lett. 34, 1347–1348 (1998).
[Crossref]

T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997).
[Crossref] [PubMed]

Koch, W.

G. von Freymanna, W. Koch, D. C. Meisel, and M. Wegener, “Diffraction properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 83, 614–616 (2003).
[Crossref]

Lederer, F.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous Refraction and Diffraction in Discrete Optical Systems,” Phys. Rev. Lett. 88, 093901–093904 (2002).
[Crossref] [PubMed]

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[Crossref]

Mandelik, D.

Martin, H.

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J.-U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic-organic hybrid materials for application in optical devices,” Thin Solid Films 422, 194–200 (2003)
[Crossref]

Meisel, D. C.

G. von Freymanna, W. Koch, D. C. Meisel, and M. Wegener, “Diffraction properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 83, 614–616 (2003).
[Crossref]

Modotto, D.

Mogilevtsev, D.

Morandotti, R.

R. Morandotti, H. S. Eisenberg, D. Mandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, and J. S. Aitchison, “Interactions of discrete solitons with structural defects,” Opt. Lett. 28, 834–836 (2003).
[Crossref] [PubMed]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Nonlinearly Induced Escape from a Defect State,” Appl. Phys. Lett. 75, 1384–1386 (1999).
[Crossref]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[Crossref]

Notomi, M.

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
[Crossref]

Park, J.-U.

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J.-U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic-organic hybrid materials for application in optical devices,” Thin Solid Films 422, 194–200 (2003)
[Crossref]

Pertsch, T.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous Refraction and Diffraction in Discrete Optical Systems,” Phys. Rev. Lett. 88, 093901–093904 (2002).
[Crossref] [PubMed]

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[Crossref]

Peschel, U.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous Refraction and Diffraction in Discrete Optical Systems,” Phys. Rev. Lett. 88, 093901–093904 (2002).
[Crossref] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[Crossref]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Nonlinearly Induced Escape from a Defect State,” Appl. Phys. Lett. 75, 1384–1386 (1999).
[Crossref]

Popall, M.

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J.-U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic-organic hybrid materials for application in optical devices,” Thin Solid Films 422, 194–200 (2003)
[Crossref]

U. Streppel, P. Dannberg, C. Wächter, A. Bräuer, L. Fröhlich, R. Houbertz, and M. Popall, “New wafer-scale fabrication method for stacked optical waveguide interconnects and 3D micro-optic structures using photo-responsive (inorganic-organic hybrid) polymers,” Opt. Mat. 21, 475–483 (2002).
[Crossref]

Russell, P

P Russell, “Photonic Crystal Fibers,” Science 299, 358–363 (2003).
[Crossref] [PubMed]

Russell, P. ST. J.

Schmitt, A.

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J.-U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic-organic hybrid materials for application in optical devices,” Thin Solid Films 422, 194–200 (2003)
[Crossref]

Silberberg, Y.

R. Morandotti, H. S. Eisenberg, D. Mandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, and J. S. Aitchison, “Interactions of discrete solitons with structural defects,” Opt. Lett. 28, 834–836 (2003).
[Crossref] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[Crossref]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Nonlinearly Induced Escape from a Defect State,” Appl. Phys. Lett. 75, 1384–1386 (1999).
[Crossref]

Somekh, S.

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–48 (1973).
[Crossref]

Sorel, M.

Stanley, C. R.

Streppel, U.

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J.-U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic-organic hybrid materials for application in optical devices,” Thin Solid Films 422, 194–200 (2003)
[Crossref]

U. Streppel, P. Dannberg, C. Wächter, A. Bräuer, L. Fröhlich, R. Houbertz, and M. Popall, “New wafer-scale fabrication method for stacked optical waveguide interconnects and 3D micro-optic structures using photo-responsive (inorganic-organic hybrid) polymers,” Opt. Mat. 21, 475–483 (2002).
[Crossref]

von Freymanna, G.

G. von Freymanna, W. Koch, D. C. Meisel, and M. Wegener, “Diffraction properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 83, 614–616 (2003).
[Crossref]

Wächter, C.

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J.-U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic-organic hybrid materials for application in optical devices,” Thin Solid Films 422, 194–200 (2003)
[Crossref]

U. Streppel, P. Dannberg, C. Wächter, A. Bräuer, L. Fröhlich, R. Houbertz, and M. Popall, “New wafer-scale fabrication method for stacked optical waveguide interconnects and 3D micro-optic structures using photo-responsive (inorganic-organic hybrid) polymers,” Opt. Mat. 21, 475–483 (2002).
[Crossref]

Wegener, M.

G. von Freymanna, W. Koch, D. C. Meisel, and M. Wegener, “Diffraction properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 83, 614–616 (2003).
[Crossref]

Yariv, A.

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–48 (1973).
[Crossref]

Zentgraf, T.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous Refraction and Diffraction in Discrete Optical Systems,” Phys. Rev. Lett. 88, 093901–093904 (2002).
[Crossref] [PubMed]

Appl. Phys. Lett. (3)

G. von Freymanna, W. Koch, D. C. Meisel, and M. Wegener, “Diffraction properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 83, 614–616 (2003).
[Crossref]

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–48 (1973).
[Crossref]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Nonlinearly Induced Escape from a Defect State,” Appl. Phys. Lett. 75, 1384–1386 (1999).
[Crossref]

Electron. Lett. (1)

J. C. Knight, T. A. Birks, R. F. Cregan, P. ST. J. Russell, and J.-P. de Sandro, “Large Mode Area Photonic Crystal Fiber,” Electron. Lett. 34, 1347–1348 (1998).
[Crossref]

Opt. Express (1)

Opt. Lett. (3)

Opt. Mat. (1)

U. Streppel, P. Dannberg, C. Wächter, A. Bräuer, L. Fröhlich, R. Houbertz, and M. Popall, “New wafer-scale fabrication method for stacked optical waveguide interconnects and 3D micro-optic structures using photo-responsive (inorganic-organic hybrid) polymers,” Opt. Mat. 21, 475–483 (2002).
[Crossref]

Phys. Rev. B (1)

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
[Crossref]

Phys. Rev. Lett. (3)

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous Refraction and Diffraction in Discrete Optical Systems,” Phys. Rev. Lett. 88, 093901–093904 (2002).
[Crossref] [PubMed]

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[Crossref]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[Crossref]

Science (1)

P Russell, “Photonic Crystal Fibers,” Science 299, 358–363 (2003).
[Crossref] [PubMed]

Thin Solid Films (1)

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J.-U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic-organic hybrid materials for application in optical devices,” Thin Solid Films 422, 194–200 (2003)
[Crossref]

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Figures (6)

Fig. 1.
Fig. 1. Cross section of a polymer waveguide array with a defect. The defect is introduced by reducing the waveguide spacing compared with the homogenous array.
Fig. 2.
Fig. 2. Dispersion relation of Bloch modes and the formation of defect modes (a) diffraction relation of Bloch-waves (longitudinal vs. transverse wavenumber) in a homogeneous waveguide array (propagation constant of isolated waveguides: β0, coupling constant C). In the shaded regions only evanescent waves exist. (b) Shift of the band structure and formation of a staggered mode (wavenumber βD) around a defect with a wavenumber reduced by δβ. (c) Expansion of the band structure and formation of staggered and unstaggered modes around a defect with increased coupling (C 1>C) (d) Compression of the band structure around a defect with reduced coupling (C 1<C).
Fig. 3.
Fig. 3. Field and intensity of a staggered and an unstaggered mode for a dominant change of the propagation constant of the defect. Field (a) and intensity (b) distribution of an unstaggered defect mode for δβ/C=2.0 and C 1/C=0.8 (cross 1 in Fig. 4), solid line: theory, dots: experiment. Field (c) and intensity (d) distribution of a staggered defect mode for δβ/C=-1.4 and C 1/C=1.1 (cross 2 in fig. 4), solid line: theory, dots: experiment.
Fig. 4.
Fig. 4. Regions of existence for symmetric staggered and unstaggered modes in the (C1/C)2-δβ-plane. The crosses mark the parameters for the experiments (Figs. 3, 5 and 6).
Fig. 5.
Fig. 5. Interference pattern of a staggered and an unstaggered defect mode for dominant change of the coupling constant (δβ/C=-0.3 and C 1/C=1.4, cross 3 in Fig. 4) of the defect at a propagation distance of 59,95mm. Dots: experiment, lines: theory, dashed line: position of the excitation. (b) Intensity distribution for an excitation of the defect waveguide. (a) and (c) Intensity distribution for an excitation of the left and right nearest neighbor waveguide of the defect. Insets: schematic diagrams of the modal amplitude of the unstaggered and staggered mode, the superposition of both modal fields produces the actual interference pattern.
Fig. 6.
Fig. 6. Diffraction pattern for an excitation of a repulsive defect ((a) theory, (b) experiment) with reduced coupling (C 1/C=0.5, δβ/C=0, cross 4 in Fig.4) and in a homogeneous array ((c) theory, (d) experiment).

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

( i Z + β 0 ) A n + C ( A n + 1 + A n 1 ) = 0 ,
β = β 0 + 2 C cos ( κ ) .
( i Z + β 0 ) A ± 1 + C 1 A 0 + C A ± 2 = 0 ,
A n ( Z ) = a n exp ( i β D Z )
a ± n = a ± 1 γ n 1 for n 2
γ < 1
β D = β 0 + C ( γ + 1 γ )
1 γ = δβ 2 C ± ( δβ 2 C ) 2 + 2 ( C 1 C ) 2 1 .

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